c Pleiades Publishing, Ltd., 2018. ISSN 0021-8944, Journal of Applied Mechanics and Technical Physics, 2018, Vol. 59, No. 1, pp. 120–131.  c V.M. Kornev, A.G. Demeshkin. Original Russian Text 

QUASI-BRITTLE FRACTURE OF COMPACT SPECIMENS WITH SHARP NOTCHES AND U-SHAPED CUTS V. M. Kornev and A. G. Demeshkin

UDC 539.3

Abstract: A two-parameter (coupled) discrete-integral criterion of fracture is proposed. It can be used to construct fracture diagrams for compact specimens with sharp cracks. Curves separating the stress–crack length plane into three domains are plotted. These domains correspond to the absence of fracture, damage accumulation in the pre-fracture region under repeated loading, and specimen fragmentation under monotonic loading. Constants used for the analytical description of fracture diagrams for quasi-brittle materials with cracks are selected with the use of approximation of the classical stress–strain diagrams for the initial material and the critical stress intensity factor. Predictions of the proposed theory are compared with experimental results on fracture of compact specimens with different radii made of polymethylmethacrylate (PMMA) and solid rubber with crack-type effects in the form of U-shaped cuts. Keywords: brittle and quasi-brittle fracture, small-scale yielding, necessary and sufficient criteria of fracture, elastoplastic material, edge crack, U-shaped cut. DOI: 10.1134/S0021894418010157

INTRODUCTION The analysis of various systems used for fracture calculations within the framework of the linear fracture mechanics (LFM) and nonlinear fracture mechanics (NLFM) [1, 2] shows that no simple schemes for calculating structures made of quasi-brittle materials are available at the moment. It is desirable to use commonly accepted characteristics of materials in calculations. Zhu and Joyce noted in their review paper [2] that the most important parameters used in fracture mechanics are J-integral, the stress intensity factor K, crack width at the tip, and angle at the crack tip. In LFM calculations of the critical stresses for specimens with sufficiently long cracks, the only governing parameter is the stress intensity factor (SIF). Berto and Lazzarin [3] considered one-parameter local criteria of fracture of brittle and quasi-brittle solids in the vicinity of stress concentrators. Below we consider elastoplastic bodies under ultimate deformation. In studying fracture of quasi-brittle bodies with small-scale yielding, one has to use quasi-linear fracture mechanics (QLFM). The proposed QLFM version involves the use of the critical SIF and the classical stress–strain diagram with allowance for the ultimate deformation of the examined material. The goals of the present work are to derive relations for the critical stresses in compact specimens with sharp cracks and to compare the theoretically predicted critical loads with experimental data obtained in the case of fracture of compact specimens with U-shaped cuts.

Lavrent’ev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia; [email protected]; [email protected]. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 59, No. 1, pp. 138–152, January–February, 2018. Original article submitted July 20, 2016; revision submitted November 18, 2016. 120

c 2018 by Pleiades Publishing, Ltd. 0021-8944/18/5901-0120 

(a) s1 y sY D= /0

_D*

0

x

s1

(b) sy sY

0 r

_D*

x

(c) v

_D*

0

x

Fig. 1. Loading configuration (a), stress field (b), and crack propagation scheme (c) in the vicinity of the model crack tip.

1. MODEL THAT DESCRIBES FRACTURE OF COMPACT SPECIMENS WITH SHARP CRACKS To consider fracture of a real material, we choose a model elastoplastic material with limited deformations. We consider a two-link approximation of the σ–ε diagram of the initial material, which has the following parameters: elasticity modulus E, conditional yield stress of the material σY , i.e., constant stresses acting on the crack continuation according to the modified Leonov–Panasyuk–Dugdale model [4, 5], maximum elastic strain of the material ε0 (σY = Eε0 ), and maximum (limiting) strain of the material ε1 . Let r be a phenomenological size for a material with a regular structure, which can be treated as a conditional grain diameter. The Neuber–Novozhilov approach [6, 7] allows solutions with a singular component to be used for structured media. A model of sharp flat crack propagation in a quasi-brittle material is proposed. Mode I fracture is considered, and the crack is modeled by a cut. Let us assume that the initial stage of crack propagation is described by the modified Leonov–Panasyuk–Dugdale model [4, 5]. The edges of the model crack are braced by the stresses σY corresponding to the conditional yield stress. 121

For the proposed model, Fig. 1 shows the loading configuration (σ∞ are the normal stresses prescribed at infinity), the stresses on the crack line near the model crack tip, and the scheme of model crack propagation in the vicinity of its tip. Let us consider a sharp edge crack of length l0 . As a sharp crack, we consider model edge cracks (cuts) of length l = l0 + Δ. The pre-fracture regions of the model cracks are located on the continuation of the axis of symmetry of the crack (l and Δ are the lengths of the model cracks and pre-fracture regions, respectively). In the case of small-scale yielding, the sufficient (coupled) criterion of fracture for the sharp crack can be presented in the following form [8]: r 1 σy (x, 0) dx = σY ; (1) r 0

2ν(−Δ∗ , 0) = δ ∗ .

(2)

Here σy (x, 0) are the normal stresses on the continuation of the model edge crack, 2ν = 2ν(x, 0) is the model crack width (x < 0), δ ∗ is the critical width of the model crack, and Δ∗ is the critical length of the pre-fracture region (the critical values obtained with the use of the sufficient and necessary fracture criteria are marked by the asterisk and zero superscripts, respectively). The x axis of the rectangular coordinate system with the origin at the model crack tip is directed along the crack, and the y axis is aligned normal to the crack plane. It should be noted that the proposed criterion involves two parameters; moreover, Eq. (1) is satisfied with allowance for averaging of stresses near the model crack tip, whereas Eq. (2) is valid at the point corresponding to the real crack tip. Thus, in formulating the sufficient criterion (1), (2), we use the integration interval 0  x  r and the point x = −Δ∗ in the pre-fracture region (see Figs. 1b and 1c). Equality (1) of the sufficient criterion (1), (2) is a typical force criterion of fracture [9], whereas equality (2) is a deformation criterion [9]. The pre-fracture region occupies some part of the plasticity region. The proposed criterion (1), (2) describes brittle fracture (Δ ≡ 0) and quasi-brittle fracture (Δ > 0) at Δ∗  l0 ,

(3)

which corresponds to small-scale yielding in the case of fracture of cracked bodies. All necessary transformations with appropriate comments for cracked structures can be found in [8, 10, 11]. The stress field in the vicinity of the model crack tips x > 0 for compact specimens can be presented in the following form [9, 12, 13]: σy (x, 0)  KI /(2πx)1/2 + σnom ,

KI = KI∞ + KIΔ ,

KI∞ > 0,

KIΔ < 0.

(4)

Here σy (x, 0) are the nominal stresses at the Ox axis, σnom are the nominal stresses, i.e., the estimates of nonsingular terms of solutions near the model crack tips, KI∞ is the SIF induced by prescribed test conditions, and KIΔ is the SIF induced by the constant stresses −σY acting in the pre-fracture region. The total SIF KI at the model crack tip is positive because small-scale yielding is considered. Figure 2 shows a compact specimen with a narrow U-shaped cut (P is the load applied in the experiment and R is the rounding radius). The sizes of the tested specimens are consistent with recommendations given in the reference books [12, 13]. Relations (4) include the singular and regular parts of the solutions. Let us first describe the regular part of the solution, using the simplest approximation of the stress field. Let the regular part of the stress field correspond to the material strength approximation [9]. The stress state in compact specimens with an edge crack is the sum of the stress states induced by tension and bending. The influence of cracks in structural elements subjected to bending on their fracture was studied in [14]. Eccentric tension of a compact specimen subjected to elastic straining was considered in [9, pp. 96, 97]. Equilibrium conditions for forces and moments are used for estimating the regular part of the solution in Eqs. (4). The diagrams of the tensile and bending stresses for the compact specimen are shown in Fig. 2. The regular parts of the solutions for the normal stresses in the vicinity of the model crack tip (x = 0) in the case of eccentric tension are expanded into the Maclaurin series. In the resultant expansion, we retain only the first term, which contains information about the constant tensile stresses and bending stresses at the point x = 0; following [6], these stresses at the point x = 0 are called the nominal stresses. For compact specimens with a sharp crack, the proposed approximation of the normal stresses at the model crack tips has the form 122

P

R x

O

l0 w P t(w_l0)

P

x

O w_l0 2 3P(w+l0) t(w_l0)2 O l0 +

x

w_l0 2

Fig. 2. Compact specimen with a narrow U-shaped cut.

σnom = σ1nom + σ2nom ,

σ1nom =

P , t(w − l)

σ2nom =

3P (w + l) . t(w − l)2

(5)

Here σ1nom and σ2nom are the nominal stresses due to tension and bending, respectively (see Fig. 2), and t is the compact specimen thickness. The first and second terms correspond to the constant tensile stresses and bending stresses at the model crack tips. As l → w, we have σnom → ∞ in Eq. (5), which corresponds to the growth of the stresses due to net-section decreasing down to zero under constant loading.

2. SIF ESTIMATES AND CRITICAL STRESSES IN COMPACT SPECIMENS As we study deformation of materials under small-scale yielding conditions, the following relation is valid for compact specimens with sharp cracks: KI = KI∞ (P/(tw), l, l/w) + KIΔ (l, Δ, σY ) > 0. The expression for the SIF KI∞ caused by the prescribed test conditions for compact specimens with a sharp crack can be presented as √ KI∞ = (P/(tw))Y (l/w) πl, (6) 2 3 4 Y (l/w) = 16.7 − 104l/w + 370(l/w) − 574(l/w) + 361(l/w) . Here the expression for the K-calibration is borrowed from the reference book [15, pp. 34, 35]. The SIF KIΔ caused by the constant stresses −σY acting in the pre-fracture region has a universal presentation. For the half-plane with the edge crack, the SIF KIΔ is calculated as follows (see [12, pp. 113–114; 13]): 123

KIΔ = −σY

√ πl [1 + f (1 − Δ/l)][1 − (2/π) arcsin (1 − Δ/l)].

(7)

Here f (1 − Δ/l) is a certain function with the following estimate: 0  f (1 − Δ/l)  0.0138 at Δ/l  0.1 [12, 13]. Inequality (3) is valid for the quasi-brittle approximation; therefore, the value of the function f (1 − Δ/l) can be neglected. The following presentation for the term arcsin (1 − Δ/l) in Eq. (7) is valid with accuracy to the highest order of smallness if quasi-brittle fracture is considered:  arcsin (1 − Δ/l)  π/2 − 2Δ/l , Δ/l  1. After appropriate simplifications, we obtain the following expression for the SIF KIΔ :  KIΔ ≈ −2σY 2Δ/π.

(8)

If there is a singular component of the solution under small-scale yielding conditions (5), the expression for the model crack opening displacement 2v in the compact specimen with a sharp crack can be presented in the following form [9, pp. 30–32]:  l  −x η+1  3−μ KI l, Δ, , KI > 0, . (9) 2v(−x, 0) ≈ ηd = 3 − 4μ, ηs = G w 2π 1+μ Here η are the coefficients ηd and ηs for the plane deformation (strain) state and plane stress state, G = E/[2(1+μ)] is the shear modulus, and μ is Poisson’s ratio. For specimens with sharp cracks, the critical opening displacement of the model cracks δ ∗ in Eq. (2) is calculated by the formula δ ∗ = (ε1 − ε0 )a. (10) The width a of the pre-fracture region for compact specimens with sharp cracks in Eq. (10) is assumed to be identical to the width of the plasticity region at the real crack tip:  [KI∞ ]2  3 2 a= . (11) + (1 − 2μ) 2π(σY )2 2 The width of the plasticity region in Eq. (11) depends on the initial crack length l0 [9], i.e., presentation (5) is used for the SIF KI∞ , where the model crack length l and the parameter l/w are replaced by the initial crack length l0 and the parameter l0 /w, respectively. Relation (11) is written for a plane strain state. In the case of a plane stress state, we have μ = 0 in this relation. The critical opening displacement of the model crack δ ∗ in Eqs. (10) and (11) corresponds to the transition of the material at the real crack tip to the critical state and its fracture. Let us estimate the parameters of the critical state of the material at the crack tip in the compact specimen. All information necessary for using the two-parameter sufficient criterion (1), (2) is contained in relations (4)–(6) and (8)–(11). After appropriate manipulations, the initial equalities of criterion (1), (2) transform to the approximate equalities for compact specimens with sharp cracks:   P ∗ 3P (w + l∗ )  πr P ∗ l ∗ ∗ ,l , − , KI∞ + KIΔ (l , Δ , σY ) ≈ σY − tw w t(w − l∗ ) t(w − l∗ )2 2 (12)  P  [K (σ , l , l /w)]2  Δ∗ 3 2(η + 1)(1 + μ)  l∗  I∞ ∞ 0 0 KI∞ , l∗ , + KIΔ (l∗ , Δ∗ , σY ) ≈ (ε1 − ε0 ) + (1 − 2μ)2 . E tw w 2π 2 2π(σY )2 It should be noted that the second equation of system (12) involves the initial crack length l0 . As we consider the quasi-brittle approximation, we replace this length l0 by the critical length of the model crack l∗ with allowance for inequality (3). After some transformations, we obtain the following system of equations:  l∗  2l∗ 2√2  2l∗  Δ∗ ∗ ∗ ∗ 1 1 + l∗ /w σ∞ σ∞ σ∞ Y ≈ 1 − − 3 , − σY w r π r l∗ 1 − l∗ /w σY (1 − l∗ /w)2 σY  l∗  2√2  Δ∗  1  Δ∗ ∗ 2(η + 1)(1 + μ)  σ∞ √ − Y E/σY σY w π l∗ l∗ 2π ≈ (ε1 − ε0 ) 124

∗ 2 3 + 2(1 − 2μ)2   l∗  σ∞ , Y 4 w σY

∗ σ∞ P∗ = . σY twσY

(13)

∗ Here σ∞ /σY are the dimensionless conditional critical stresses in compact specimens corresponding to the sufficient fracture criterion and P ∗ is the critical load. After opening the square brackets in the left side of the second equation of system (13), there arises a term with a factor Δ∗ /l∗ , which is  neglected in what follows by virtue of inequality (3). The transformed system (13) contains terms with the factors Δ∗ /l∗ . Solving the simplified system, we obtain ∗ the analytical expressions for the dimensionless conditional critical stresses σ∞ /σY and normalized critical lengths ∗ ∗ Δ /l of the pre-fracture regions:   ∗ σ∞ 3(1 + l∗ /w)  1 3 + 2(1 − 2μ)2 ε1 − ε0   l∗  2l∗ −1 Y + ≈ + 1− , σY 1 − l∗ /w (1 − l∗ /w)2 8π(1 − μ2 ) ε0 w r

 l∗ 2 ∗ [3 + 2(1 − 2μ)2 ]2  ε1 − ε0 2  σ∞ Δ∗ ≈ Y ; l∗ 29 (1 − μ2 )2 ε0 σY w 3 + 2(1 − 2μ)2 ε1 − ε0 < 1. 8π(1 − μ2 ) ε0

(14)

(15)

Inequality (15) is a constraint on the existence of quasi-brittle fracture under the condition of small-scale yielding in the pre-fracture region. ∗ Relations (14) and constraint (15) for the critical parameters σ∞ /σY and Δ∗ /l∗ are valid in the case of a plane strain state of the specimen. In the case of a plane stress state, one has to assume that μ = 0 in Eqs. (14). The presentation of the critical parameters (14) for the sufficient fracture criterion is similar to formulas derived in [8, 10, 11, 14, 16] for the corresponding parameters of specimens of another type. As ε1 → ε0 , the first equation of (14) yields a formula corresponding to the necessary criterion of fracture. 0 The conditional critical stresses σ∞ /σY are calculated by the formula   l   2l −1 0 σ∞ 3(1 + l0 /w) 1 0 0 + ≈ +Y . (16) σY 1 − l0 /w (1 − l0 /w)2 w r 0 ∗ Relation (16) describes brittle fracture of materials. Obviously, σ∞ /σY < σ∞ /σY for l0 < l∗ . Relations (14) and (16) involve a parameter r, which characterizes the effective diameter of the fracture structures. If the critical values of the SIF KIc are known, then the following presentations are valid for the effective diameters of the fracture structures of brittle (r0 ) and quasi-brittle (r) materials [16]:

r0 =

2  KIc 2 , π σY

r=

3 + 2(1 − 2μ)2 ε1 − ε0 2 2  KIc 2  1− . π σY 8π(1 − μ2 ) ε0

Thus, if the critical SIF KIc and the classical σ–ε diagram (more exactly, its approximation) are obtained in two laboratory experiments, then one can use three parameters (r, σY , and (ε1 − ε0 )/ε0 ) for constructing two ∗ 0 critical curves σ∞ /σY and σ∞ /σY in the stress–crack length plane for cracks whose length varies in a wide range, with Poisson’s ratio μ being taken into account. The constructed curves (diagrams of quasi-brittle fracture for specimens of the type considered in the present study) depend on the geometric ratio l∗ /w, which characterizes the chosen specimen type. Two curves separate the stress–crack length plane into three domains corresponding to the absence of fracture, damage accumulation in the material in the pre-fracture region under repeated loading, and specimen fragmentation under monotonic loading. 0 Figure 3 shows the dimensionless conditional critical stresses σ∞ [2l0 /r, l0 /w]/σY (curves 1 and 3) and ∗ ∗ ∗ σ∞ [2l /r, (ε1 − ε0 )/ε0 , l /w]/σY (curves 2 and 4) for compact specimens with sharp cracks. The pairs of curves 1, 2 and 3, 4 are diagrams of quasi-brittle fracture for the specimen type considered in the study. In the present calculations by Eqs. (14) and (16), we used the parameters l0 /w = 0.2 and 0.4 for curves 1 and 3 and the parameters l∗ /w = 0.2 and 0.4 for curves 2 and 4, respectively. We also assumed that (ε1 − ε0 )/ε0 = 1.5 and μ = 0 because Poisson’s ratio produces a minor effect on the critical parameters. Domains located between curves 1, 2 and 3, 4 are of greatest interest because they are domains of material damage accumulation in the pre-fracture region in the case of pulsed loading with subsequent unloading [17].

125

0 /s s * /s s1 Y. 1 Y

0.10 3

0.07 0.05

2

4 1

0.03 0.02

0.01 1

102

10

103 2l/r

0 ∗ Fig. 3. Conditional critical stresses σ∞ /σY (1, 3) and σ∞ /σY (2, 4) in compact specimens with sharp cracks: l0 /w = 0.2 (1), l∗ /w = 0.2 (2), l0 /w = 0.4 (3), and l∗ /w = 0.4 (4).

Table 1. Basic parameters of approximation for PMMA specimens Specimen number

E, MPa

ε0 , %

ε1 , %

σY , MPa

1 2

3155.2 3133.9

1.23 1.51

2.41 2.18

38.80 47.32

Table 2. Basic parameters of approximation for solid rubber specimens Specimen number

E, MPa

ε0 , %

ε1 , %

σY , MPa

1 2 3

2980 3334 2401

1.63 1.29 2.14

3.63 3.20 3.15

48.57 43.06 51.49

3. EXPERIMENTAL ESTIMATES OF THE CRITICAL STATES OF COMPACT SPECIMENS WITH U-SHAPED CUTS Experiments with compact specimens containing U-shaped cuts were performed; crack defects were modeled by narrow notches. The specimens were made of polymethylmethacrylate (PMMA) and solid rubber. These material exhibit a quasi-brittle behavior during fracture [3, 11, 14–18]. The initial characteristics of the materials used in the study are the critical SIF KIc and the classical σ–ε diagram (more exactly, its two-link approximation). The initial σ–ε diagram was approximated by the two-link σ–ε diagram in such a way that the areas of the domain bounded by the corresponding curves σ = σ(ε), straight line ε = ε1 , and abscissa axis were identical. Using the results obtained in two preliminary conducted laboratory experiments, we estimated the parameters r, σY , and (ε1 −ε0 )/ε0 of the considered materials. Using these parameters ∗ 0 /σY and σ∞ /σY for specimens and formulas (14) and (16), we then calculated the conditional critical stresses σ∞ ∗ 0 with sharp cracks. Then, the theoretically predicted values of the stresses σ∞ /σY and σ∞ /σY were compared with experimental data on fracture of compact specimens with U-shaped cuts with known radii R of these cuts. Figures 4 and 5 and Tables 1 and 2 show the experimental data used as a basis for fitting the classical σ–ε diagrams of the considered materials by two-link diagrams. The variables listed in Table 1 for PMMA specimens had the following mean values: E = 3145 MPa, ε0 = 1.37%, ε1 = 2.30%, and σY = 43.1 MPa. The variables listed 126

s, MPa 60

s, MPa 60 2

3 4

50

50 3

40

30

20

20

10

10 0.5

1.0

1.5

2.5 e, %

2.0

1

40

30

0

2

1

0

1

Fig. 4.

2

4 e, %

3

Fig. 5.

Fig. 4. Classical σ–ε diagram for PMMA: curves 1 and 2 are the results for the specimens denoted by the corresponding numbers; curve 3 is the result of approximation by the proposed method. Fig. 5. Classical σ–ε diagram for solid rubber: curves 1–3 show the results for the specimens denoted by the corresponding numbers; curve 4 is the result of approximation by the proposed method.

Table 3. Averaged parameters of quasi-brittle materials r, mm

2l0 /r

Material

KIc , MPa · m1/2

σY , MPa

ε1 − ε0 ε0

μ = 0

μ=0

μ = 0

μ=0

PMMA Solid rubber

1.0 1.3

43 48

0.68 0.97

0.279 0.344

0.258 0.304

71.7 58.1

77.5 65.8

in Table 2 for solid rubber specimens had the following mean values: E = 2905 MPa, ε0 = 1.69%, ε1 = 3.33%, and σY = 47.7 MPa. The experiments were performed on the Zwick/Roell TC-FR 100TL A4K universal testing machine. The averaged parameters of quasi-brittle materials (PMMA and solid rubber) are listed in Table 3. The following estimates of the critical SIF KIc for the PMMA specimen were obtained: KIc = (1.02 ± 0.05) MPa · m1/2 [15, p. 161] and KIc ≈ 1 MPa · m1/2 [18, p. 157]. The critical values of KIc are in good agreement. The values obtained in the laboratory experiments aimed at determining the critical SIF KIc for PMMA almost coincide with KIc . In the calculations described below, we used the value KIc = 1 MPa · m1/2 . It should be noted that the conditional yield stress σY for PMMA and solid rubber specimens obtained by using the proposed approximations of the σ–ε diagrams is appreciably greater than the real yield stress σ0.2 . The parameter (ε1 − ε0 )/ε0 characterizing the quasi-brittleness of the tested materials satisfies constraint (15). It follows from the calculated results that Poisson’s ratio μ = 0.4 or μ = 0 exerts a minor effect on the parameter r for PMMA and solid rubber. The length l0 = 10 mm was used for calculating the normalized lengths 2l0 /r of crack-type defects in compact specimens with U-shaped cuts. The stress state of the laboratory specimens is an intermediate state between the plane strain state (μ = 0) and plane stress state (μ = 0) if the three-axis character of the stress state is taken into account. Figure 6 shows the fractured compact specimens of two types with U-shaped cuts. Tables 4 and 5 illustrate the experimental results on fracture of two types of compact specimens of width w = 50 mm with U-shaped cuts [P are the fracture loads of specimens made of PMMA sheets of thickness t = 6.3 mm and solid rubber sheets of ∗ thickness t = 7 mm, P¯ are the mean values of the fracture loads, and σ ¯∞ /σY are the mean dimensionless values of ∗ ¯ the fracture loads calculated by the formula σ ¯∞ /σY = P /(twσY ) in Eqs. (13)]. The scatter of the values of the fracture loads for PMMA and solid rubber is caused by a significant scatter of the maximum strain values ε1 , which affects the conditional yield stress of the material σY in the case of using the proposed approximation. 127

(a)

(b)

Fig. 6. Fractured compact specimens made of PMMA (a) and solid rubber (b) with U-shaped cuts. ∗ The mean dimensional values of the fracture stresses σ ¯∞ /σY listed in Tables 4 and 5 are compared with similar results obtained on the basis of theoretical predictions (Fig. 7). Figure 7 shows three curves for compact 0 ∗ specimens with sharp cracks: critical stresses σ∞ /σY (curve 1 for PMMA and solid rubber) and σ∞ /σY (curve 2 for PMMA and curve 3 for solid rubber), which describe brittle and quasi-brittle fracture, respectively. Relations (14) and (16) for compact specimens are used for constructing curves 2 and 3, and 1. The theoretical loads are calculated on the basis of the averaged parameters of quasi-brittle materials (see Table 3) and the initial length of the sharp crack l0 = 10 mm (it is assumed that μ = 0). As the stress state of the laboratory specimens is an intermediate state between the plane strain state and plane stress state, the mean dimensionless fracture ∗ stresses σ ¯∞ /σY correspond to the ranges of the dimensionless lengths 2l0 /r of crack-type defects given in Table 3. This interval is 58.1 < 2l0 /r < 65.8 for solid rubber (intervals I in Fig. 7) and 71.7 < 2l0 /r < 77.5 for PMMA (intervals II). Each of the six intervals for PMMA and solid rubber corresponds to the dimensionless values of the ∗ fracture stresses σ ¯∞ /σY for compact specimens with U-shaped cuts with R = 0.6, 1.0, 2.0, 3.0, 4.0, and 5.0 mm. ∗ In view of the scatter of the fracture loads P , we obtain a certain region on the plane σ∞ /σY –2l/r. The fracture (critical) loads for compact specimens with sharp cracks and U-shaped cuts are significantly different. The results of comparisons of the theoretical critical stresses for compact specimens with sharp cracks and the critical stresses for compact specimens with U-shaped cuts allow us to conclude that the frequently used processing technology of drilling out the tip of the sharp crack is fairly effective. The following constraint should be satisfied in using the drill-out technology: the radius of the U-shaped cut cannot be smaller than the pre-fracture region length Δ∗ /l∗ determined by Eq. (14) if the drill axis passes through the tip of the real crack. The parameter (ε1 − ε0 )/ε0 ∗ exerts a significant effect on the critical stresses σ ¯∞ /σY for compact specimens with U-shaped cuts made of PMMA and solid rubber.

4. DISCUSSION OF RESULTS Comparisons of the proposed model [8, 10, 11, 14, 16, 17] with the Leonov–Panasyuk–Dugdale model [4, 5] and with the Barenblatt cohesion model [19] shows that solutions obtained by using both classical models [4, 5, 19] have no singular components (hypothesis put forward by Khristianovich). As the solution obtained by using the proposed model has a singular component, it is possible to use a two-parameter (coupled) criterion. The parabolic shape of the model crack opening is caused by the presence of a singularity. In constructing the quasi-brittle fracture diagrams, we use the same results of laboratory experiments as those for determining the critical SIF KIc and for ∗ constructing the classical σ–ε diagram for the specimen material. One of the curves σ∞ /σY of the constructed diagram of quasi-brittle fracture is similar to the critical fracture diagrams for a plane with cracks of different types ∗ 0 [20, pp. 231–235]. The domain between the curves σ∞ /σY and σ∞ /σY of the proposed diagram in the stress–crack length plane is a domain with material damage in the pre-fracture region. It should be noted that the pre-fracture region length is calculated from the solution of the problem, and the pre-fracture region is some part of the plasticity 128

Table 4. Fracture loads for compact PMMA specimens with crack-type defects for t = 6.3 mm and l0 = 10 mm R, mm 0,6 1.0 2.0 3.0 4.0 5.0

P, N experiment 1

experiment 2

experiment 3

746 785 1050 1354 1825 2080

903 942 1099 1275 1962 2021

716 785 1177 1344 1766 2021

P¯ , N

∗ /σ σ ¯∞ Y

788 837 1109 1324 1851 2041

0.0582 0.0618 0.0819 0.0977 0.1367 0.1507

Table 5. Fracture loads for compact solid rubber specimens with crack-type defects for t = 7 mm and l0 = 10 mm R, mm 0.6 1.0 2.0 3.0 4.0 5.0

P, N experiment 1

experiment 2

experiment 3

824 1050 1246 1501 1579 1785

814 942 1246 1442 1589 1678

845 1059 1334 1432 1521 1756

P¯ , N

∗ /σ σ ¯∞ Y

828 1017 1275 1458 1563 1740

0.0493 0.0605 0.0759 0.0868 0.0930 0.1036

II

0 /s , s* /s s1 Y 1 Y

I 0.10 0.07 2

0.05

1

3

0.03 0.02 0.015 10-1

1

10

102

2l/r

Fig. 7. Theoretical (1–3) and experimental (I, II) critical stresses for specimens with sharp cracks 0 for PMMA and solid ruband fracture stresses for compact specimens with U-shaped cuts: σ∞ ∗ ∗ ber (1), σ∞ for PMMA (2), and σ∞ for solid rubber (3); the intervals of the values of 2l/r corresponding to the critical stresses for PMMA and solid rubber are indicated by I and II; the rounding radius is R = 0.6, 1.0, 2.0, 3.0, and 5.0 mm (the lower interval corresponds to R = 0.6 mm).

129

region. As the pre-fracture region width equal to the plasticity region width was used to construct the solution, we considered the fracture of a bimaterial formed after welding of two materials [21–26]. If the crack passed along the ∗ interface between the welded joint and the base material, the critical loads σ∞ /σY of the welded joints turned out to be appreciably lower than the corresponding critical loads for the base material (the one-sided region of plasticity is located in the weaker material). The classical models do not describe damage accumulation and fracture of welded joints. The numerical experiments [8, 22, 25, 27] showed that the proposed model ensures a sufficiently accurate description of quasi-brittle fracture, and the results calculated by using the proposed model agree well with the results predicted by the classical models for materials with a quasi-ductile type of fracture (constraint (15) is not satisfied for this type of fracture). The constructed diagrams of quasi-brittle fracture were used to describe the motion of the crack tips under fatigue loading of homogeneous [17, 28–30] and welded [31, 32] structures. It should be noted that the quasi-brittle materials in [17] were obtained from D16 and D16T structural materials after preliminary inelastic deformation (asdelivered D16 and D16T materials exhibit a quasi-ductile type of fracture). The classical models cannot describe the fatigue modes of fracture of cracked bodies. The empirical laws of fatigue fracture were described in [33, 34].

CONCLUSIONS The diagram of quasi-brittle fracture for compact specimens was constructed in the load–crack length plane. The diagram consists of two curves, which separate the plane into three domains. In the first domain, the fracture is absent. Damage accumulation under repeated loading occurs in the second domain. Finally, fragmentation of the specimen under monotonic loading occurs in the third domain. The constants for the analytical description of the fracture diagrams of quasi-brittle materials with cracks were chosen by using the approximation of the classical stress–strain diagram for the initial material and the critical stress intensity factor. Experimental results on fracture of compact specimens with crack-type defects in the form of U-shaped cuts were compared with results predicted by the proposed model. This comparison of the theoretical and experimental results revealed a significant effect of the rounding radius on the fracture load value. Recommendations are given on choosing the rounding radius of the U-shaped cut along the pre-fracture region for the case of drilling out the damaged material at the tip of a sharp crack. This work was supported by the Russian Foundation for Basic Research (Grant Nos. 16-08-00483 and 18-08-00528).

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